Objects moving together in object mode
It sounds like you probably have at least one (or maybe more) unwanted mirror modifiers on your eyes, at least one of which is set up to mirror using another object in your scene.  When you select the body and the eyes and try to move them, you're seeing the mirror reflection of the eyes move away from the body because they're mirrored across some other object that isn't being moved.  The original eyes might even be inside the body where you don't realize they're (correctly) moving with the body.

I'd suggest that you check the set of modifiers on your eyes and delete any mirrors.  If your eyes go missing entirely, you might want to select your body and "h" to hide it, to see if the eyes are actually inside.  (Afterwards, Alt-h will unhide anything you've hidden.)

If you end up with only one eye and want it mirrored, there are a few ways to do it right.  One way is:


Set the origins of both the body and the single eye to their respective geometries.
Position the eye in one socket.
Add a Mirror modifier with "Mirror Object" set to the body (so it's mirroring across the body origin), and check a single, appropriate axis (e.g., just "X") to mirror the eye in the right direction.


Then, shift-selecting both the eye(s) and the body should allow you to move them together.  You can also parent the eye(s) to the body and moving the body should automatically move the eyes.